Average Error: 43.6 → 0.5
Time: 6.4s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1 \cdot \frac{4}{\frac{-b}{a \cdot c} - \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot c}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1 \cdot \frac{4}{\frac{-b}{a \cdot c} - \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot c}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r39036 = b;
        double r39037 = -r39036;
        double r39038 = r39036 * r39036;
        double r39039 = 4.0;
        double r39040 = a;
        double r39041 = r39039 * r39040;
        double r39042 = c;
        double r39043 = r39041 * r39042;
        double r39044 = r39038 - r39043;
        double r39045 = sqrt(r39044);
        double r39046 = r39037 + r39045;
        double r39047 = 2.0;
        double r39048 = r39047 * r39040;
        double r39049 = r39046 / r39048;
        return r39049;
}
↓
double f(double a, double b, double c) {
        double r39050 = 1.0;
        double r39051 = 4.0;
        double r39052 = b;
        double r39053 = -r39052;
        double r39054 = a;
        double r39055 = c;
        double r39056 = r39054 * r39055;
        double r39057 = r39053 / r39056;
        double r39058 = r39052 * r39052;
        double r39059 = r39051 * r39054;
        double r39060 = r39059 * r39055;
        double r39061 = r39058 - r39060;
        double r39062 = sqrt(r39061);
        double r39063 = r39062 / r39056;
        double r39064 = r39057 - r39063;
        double r39065 = r39051 / r39064;
        double r39066 = r39050 * r39065;
        double r39067 = 2.0;
        double r39068 = r39067 * r39054;
        double r39069 = r39066 / r39068;
        return r39069;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 43.6

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
  2. Using strategy rm

  3. Applied flip-+43.7

    \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  4. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

  7. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

  8. Applied times-frac0.4

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  9. Simplified0.4

    \[\leadsto \frac{\color{blue}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]

  10. Simplified0.5

    \[\leadsto \frac{1 \cdot \color{blue}{\frac{4}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot c}}}}{2 \cdot a}\]
  11. Using strategy rm

  12. Applied div-sub0.5

    \[\leadsto \frac{1 \cdot \frac{4}{\color{blue}{\frac{-b}{a \cdot c} - \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot c}}}}{2 \cdot a}\]

  13. Final simplification0.5

    \[\leadsto \frac{1 \cdot \frac{4}{\frac{-b}{a \cdot c} - \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot c}}}{2 \cdot a}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))